Knights could kill other knights. No rules against it. Knights could be ransomed, so they were often worth more alive than dead. Ransom didn't have to be paid, but it was possible, of course.
No rules of engagement for foot soldiers vs knights. It may have been frowned upon, but certainly not illegal or anything like that.
King Richard III died on the battle field. He was on his horse when his horse was stuck in swampy ground, and was assaulted by a group. I vaguely recall hearing that it was blows struck by polearms, which makes sense since he was still on his horse. He was struck in the head repeatedly. He's been in the news recently and his skeleton confirmed 8 separate blows to the head. Chances are some, or all, of those blows came from common man-at-arms / halberdiers / etc.